Surface maps into orbifolds Alden Walker October 22, 2013 Surface - - PowerPoint PPT Presentation

Surface maps into orbifolds

Alden Walker October 22, 2013

Surface maps

Given a homologically trivial loop γ in a topological space X, there is a surface S with boundary and a map f : S → X such that f (∂S) = γ. Problem: Find such a surface with genus(S) minimal.

γ

There are two obvious surface maps whose boundary goes to γ. One of them is better (genus 1) than the other (genus 2).

Efficient surface maps

Better problem: allow S to have multiple boundary components, where f (∂S) covers γ with arbitrary degree, and find a most efficient surface map.

Definition

A surface map f : S → X is admissible for the loop γ if S

f

− − − − → X

• 



• 

γ ∂S − − − − →

∂f

S1

• commutes. The total degree on the boundary is n(S, f ), i.e.

∂f∗([∂S]) = n(S, f )

• S1

. Then scl(γ) = inf

(S,f )

−χ(S) 2n(S, f ) where the inf is taken over all admissible surface maps.

Efficient surface maps in groups

Definition 1: We conflate X and π1(X), so if g ∈ G, then scl(g) = scl(γ) where γ represents g in K(G, 1). Definition 2: For g ∈ [G, G] we define the commutator length cl(g) to be the minimum number of commutators in a product which equals g. We define the stable commutator length scl(g) = lim

n→∞

cl(gn) n .

Lemma (Calegari)

These definitions are equivalent.

Things to notice about scl

Recall: scl(γ) = inf

(S,f )

−χ(S) 2n(S, f ) Note:

◮ The inf might not be realized (there might not exist a surface

map f : S → X such that −χ(S)/2n realizes the infimum). If there is such a surface map, it is extremal.

◮ scl seems difficult to compute. ◮ scl is inherently interesting, but it’s also connected (dual) to

bounded cohomology and quasimorphisms.

◮ γ need not be a single homologically trivial loop. It could be a

collection of loops which are together homologically trivial.

Groups in which we can compute scl

• 1. Free groups (Calegari scallop)
• 2. Free products of free abelian groups (Calegari sails)
• 3. Free abelian groups amalgamated along Zk subgroups (Tim

Susse, algorithm descended from sails) (includes torus knot complements).

• 4. Baumslag-Solitar groups (some elements) (Matt Clay, Max

Forester and Joel Louwsma)

• 5. Free products of cyclic groups (W. scylla, descended from

scallop) (surfaces with boundaries and orbifold points). Theoretically, (5) can be accomplished using (1), but this is infeasible and opaque. scylla is polynomial time and intuitive.

How to compute scl

All these scl algorithms do the following: they

• 1. Parameterize the space of surface maps admissible for a given

loop γ

• 2. in such a way that −χ/2n can be computed.

How to compute scl

All these scl algorithms do the following: they

• 1. Parameterize the space of surface maps admissible for a given

loop γ

• 2. in such a way that −χ/2n can be computed.

Example

◮ For all the algorithms, the space of surface maps is

parameterized as a polyhedron in a vector space, and the function −χ/2n is linear (scallop, scylla) or piecewise linear (sails).

◮ A consequence of these parameterizations is that scl is

rational in these groups

◮ In scallop and scylla, the parameterizations prove that

extremal surfaces exist in free groups and free products of cyclic groups.

Outline

• 1. Intro to scl computation (done)
• 2. Computing scl in free groups (scallop)
• 3. Computing scl in free products of cyclic groups (scylla)
• 4. Applications of scylla

Surface maps into free groups

How can a surface map into a free group?

a A b B a b

By making it skinny.

Labeled fatgraphs

The name for a skinny surface map is a labeled fatgraph.

a A b B b B A a

Technically, a labeled fatgraph is a graph in which every vertex has a cyclic order on the incoming edges, and every edge is labeled by a generator on one side and its inverse on the other (A = a−1) The combinatorial data of a labeled fatgraph induces the surface map.

Labeled fatgraphs

Another example:

A a B b a A b B a b

This surface map has boundary abAABB + ab.

Lemma (Culler)

Every surface map factors through a labeled fatgraph (possibly after compression).

Comparing surface maps

A a B b a A b B χ = −2, n = 1, −χ(S)/2n = 1 a A A a b B a A A a b B b B b B A a b B A a b B χ = −4, n = 3, −χ(S)/2n = 2

3

Both surfaces are admissible for abAABB + ab. The right surface is more efficient (actually, it is extremal).

Parameterizing surface maps

Any labeled fatgraph (e.g. for abAABB + ab) can be broken into pieces: rectangles, and polygons: Every rectangle is labeled on each side by a particular letter from abAABB + ab, so there are only finitely many kinds of rectangles. But we need to understand the junctions, which we’ll call polygons.

Polygons

Any polygon can be cut into triangles, and there are only finitely many kinds of triangles.

Polygons

a1 A3 A4 a7 b8 B6 a1 A4 B5 b2 A3 a1 b2 B6 a1 A3 A4 a7 b8 B6 a1 A4 B5 b2 A3 a1 b2 B6

Here is a hypothetical polygon in a fatgraph with boundary a1b2A3A4B5B6 + a7b8. We can record the triangles by recording the incoming and outgoing labels. For example, the lower left triangle could be denoted ((a1, b2), (A4, B5), (b2, A3)).

Building efficient surfaces

For a given boundary, the set of all pieces (all possible rectangles and all possible triangles in a fatgraph with the given boundary) is

• finite. Every rectangle has two edges to glue, and every triangle

has three.

A3 a1 a1 b2 A4 B5 b2 A3 a1 b2 A4 B5 b2 A3 B6 a1 a1 b2 b2 A3

Given two pieces, we can glue them together if they have compatible edges.

Building efficient surfaces

Let V be the vector space spanned by the set of all possible pieces for a boundary γ.

◮ Every fatgraph with boundary γ gives a vector in V . ◮ Given a vector in V satisfying the (linear) condition that it

can be glued up, then any way of gluing it up has boundary γ.

◮ n(S, f ) and −χ(S) are linear functions

I.e. there is a polyhedron in V which parameterizes the space of admissible surface maps for γ, and computing scl reduces to a linear programming problem.

Parameterizing surface maps into orbifolds (scylla)

In the case of a free product of cyclic groups, we repeat the same idea, but we need a new kind of piece in our labeled fatgraphs: group polygons.

a b c a A c c c b B a A c c c b B A a B b

Here is a surface map into a ∗ b ∗ c/3c with boundary acacBcAba + bBABcAcbc. Note the boundaries of the group polygons map to c3, so they bound disks.

Parameterizing surface maps into orbifolds

Lemma (W.)

Every surface map into a free product of cyclic groups factors through a labeled fatgraph, possibly after compression.

Parameterizing surface maps into orbifolds

Given a loop γ, we could enumerate all triangles, rectangles, and group polygons and do linear programming in the vector space spanned by them. But this is completely infeasible: the number of group polygons is exponential in the orders of the finite factors.

Group teeth

As a general principle, chopping a surface into smaller bits makes the linear programming problem smaller at the expense of making it trickier to reconstruct χ.

a2 a1 a4 a2 a5 1 2 3 4 gt(a2, a1, 0, a2) gt(a1, a4, 1, a2) gt(a4, a2, 2, a2) gt(a2, a5, 3, a2) gt(a5, a2, 4, a2)

In our case, we can chop the group polygons into group teeth.

Group teeth

Each group tooth remembers the adjacent letters, where it is in the group polygon, and which letter is at the bottom. The result is:

◮ There are at most o|γ|3 group teeth (not exponentially many)

in a finite factor with order o.

◮ There are linear constraints which ensure that a collection of

group teeth can be glued up into group polygons.

◮ With K group teeth, we produce K/o group polygons

(topologically, disks), so we can compute χ.

◮ Therefore, the space of admissible surfaces for γ is

parameterized by a polyhedron in a vector space (spanned by triangles, rectangles, and group teeth), and scl can be computed by linear programming.

Some surfaces in orbifolds

a b a b a b a b a b a b a a a a a a b b b b

These surfaces have boundary 6ab and 2abaab (= 2abAB) in G = a/3a ∗ b/2b. They show that sclG(ab) ≤ 1/12 and sclG(abAB) ≤ 0, and they are actually both extremal.

Patterns

Let G(n, m) = a/na ∗ b/mb, where G(∞, ∞) = a ∗ b is a free group. Given γ homologically trivial in G(∞, ∞), we can compute sclG(n,m)(γ) for any n and m.

Lemma (W.)

sclG(∞,∞)(γ) − |γ| 2(n + m) ≤ sclG(n,m)(γ) ≤ sclG(∞,∞)(γ)

Proof.

scylla parameterization. In particular, as n, m → ∞, sclG(n,m)(γ) → sclG(∞,∞)(γ).

Patterns

As n, m → ∞, sclG(n,m)(γ) → sclG(∞,∞)(γ). But how does sclG(n,m)(γ) vary in n and m? Experimentally, with interesting periodic behavior!

5 10 15 20 n 0.1 0.2 0.3 0.4 scl

The value of sclG(n,m)(aabABAAbaB) as n varies (there is no dependence on m).

Patterns

10 20 30 40 50 1.01 1.02 1.03 m scl

The value of sclG(100,m)(aba2b2a3b3A6B6) as m varies.

Immersions

The scylla parameterization is also useful for understanding immersions: if the target orbifold X is hyperbolic, then the scylla combinatorialization of a surface map f : S → X detects whether f is homotopic to an immersion with geodesic boundary. For a loop γ ∈ X, we say it virtually bounds an immersed surface if there is an immersed surface with geodesic boundary γn.

Theorem (W.)

Let X be a hyperbolic orbifold with one boundary component b (and a condition on the gcd of the orbifold point orders). If w ∈ π1(X) is any homologically trivial loop in X, then wbN virtually bounds an immersed surface for any sufficiently large N.

• cf. Calegari and Calegari-Louwsma.